We consider the problem of learning uncertainty regions for parameter estimation problems. The regions are ellipsoids that minimize the average volumes subject to a prescribed coverage probability. As expected, under the assumption of jointly Gaussian data, we prove that the optimal ellipsoid is centered around the conditional mean and shaped as the conditional covariance matrix. In more practical cases, we propose a differentiable optimization approach for approximately computing the optimal ellipsoids using a neural network with proper calibration. Compared to existing methods, our network requires less storage and less computations in inference time, leading to accurate yet smaller ellipsoids. We demonstrate these advantages on four real-world localization datasets.

翻译：我们研究参数估计问题中不确定性区域的学习问题。该区域为椭球，在满足指定覆盖概率的条件下最小化平均体积。与预期一致，在联合高斯数据假设下，我们证明最优椭球以条件期望为中心，形状由条件协方差矩阵决定。针对更实际情况，我们提出一种可微优化方法，通过具有适当校准的神经网络近似计算最优椭球。与现有方法相比，我们的网络在推理时所需的存储和计算量更少，从而能生成更精确且更小的椭球。我们通过四个真实定位数据集展示了这些优势。